The Poisson kernel, , is defined as
This is the real part of a holomorphic function. Now, what are the properties of that holomorphic function?
We define the Herglotz kernel as
where .
As (this is why the Poisson kernel solves the Dirichlet problem in the disc.) Similarly, as , some unknown function. This is the conjugate function of f, and convolution with is known as the Hilbert transform.

Now, we can write explicitly as
similar to the Poisson kernel, except for the sign function. Summing this series, while
This conjugate function of need not be bounded, even if is. In other words, the Hilbert transform is not bounded in the norm. It’s not bounded in the norm either. But it’s clearly bounded in the norm.

The Hilbert transform can be shown to be bounded for all , by first showing that it satisfies a weak-type inequality and then showing that all linear operators satisfying a weak-type inequality are bounded in (this is called the Marcinkiewicz Interpolation Theorem.) I might type that up another time.

If we let , then is sort of an envelope for , larger in absolute value and smoother, and having the property that it stretches and shrinks with the function.